A MODIFIED LEAST ACTION PRINCIPLE ALLOWING MASS
CONCENTRATIONS FOR THE EARLY UNIVERSE
RECONSTRUCTION PROBLEM
YANN BRENIER
Abstract
We address the early universe reconstruction (EUR) problem (as considered by Frisch
and coauthors in [24]), and the related Zeldovich approximate model [39]. By substituting the fully nonlinear Monge-Ampère equation for the linear Poisson equation to
model gravitation, we introduce a modified mathematical model (”Monge-Ampère gravitation/MAG”), for which the Zeldovich approximation becomes exact. The MAG model
enjoys a least action principle in which we can input mass concentration effects in a
canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developped by Ghoussoub [27]. A
fully discrete algorithm is introduced for the EUR problem in one space dimension.
Introduction
This paper addresses the early universe reconstruction (EUR) problem discussed by
Frisch and coauthors in [24, 17]. We refer to these papers for the detailed physical background of this important problem in cosmology. Here is a simplified mathematical formulation.
We consider (for simplicity) a smooth closed bounded 3D domain D ⊂ R3 and denote
the space variable by x ∈ D. We are given two times t1 > t0 > 0, two probability measures ρ0 (dx), ρ1 (dx) on D. We look for a time-dependent family of probability measures
ρ(t, dx) on D (depending continuously on the time variable t, with respect to the weak
convergence of measures), interpolating ρ0 and ρ1 , at t = t0 and t = t1 respectively, and
minimizing the following action
Z t1
Z
(0.1)
dt
α(t)2 ρ(t, dx)|v(t, x)|2 + β(t)2 |∇ϕ(t, x)|2 dx ,
t0
D
where v = v(t, x) ∈ R3 is a vector-field and ϕ = ϕ(t, x) a scalar field, respectively subject
to:
(0.2)
∂t ρ + ∇ · (ρv) = 0 ,
ρ = 1 + ∆ϕ .
Here ∆ = ∇2 is the Laplace operator. In the definition of the action, α and β are timedependent scaling parameters related to general relativity (GR). Following [24, 17] (case
of an Einstein-de Sitter universe), we set:
p
(0.3)
α(t) = t3/4 , β(t) = t−1/4 3/2 .
1
2
The formal optimality conditions read
(0.4)
v = v(t, x) =
∇θ(t, x)
,
α(t)2
∂t θ +
|∇θ|2
+ β 2ϕ = 0 ,
2α2
which can be also (still formally) written:
(0.5)
or
(0.6)
∂t (α2 v) + α2 (v · ∇)v = −β 2 ∇ϕ ,
∇×v =0,
∂t (α2 ρv) + ∇ · (α2 ρv ⊗ v) = −β 2 ρ∇ϕ ,
∇×v =0.
These equations, namely (0.2,0.6) are (up to the scaling factors α, β which come from
general relativity) nothing but the Euler-Poisson equations for a pressure-less, curl-free,
self-gravitating gas subject to classical Newton gravitation. These equations can also be
written, using ”material coordinates” ,
(0.7)
∂t (α(t)2 ∂t X(t, a)) = −β(t)2 (∇ϕ)(t, X(t, a)),
where a denotes the material coordinate, X(t, a) the position at time t of the mass particle
with label a. In the case of coefficients (0.3), we find
2t 2
∂ X(t, a) + ∂t X(t, a) = −(∇ϕ̃)(t, X(t, a)),
(0.8)
3 tt
where ϕ̃(t, x) = ϕ(t, x)/t satisfies
ρ = 1 + t∆ϕ̃.
Remarkably enough, at early stage t << 1, the density field must be uniformly equal
to 1 (otherwise solutions are unbounded) and, even more surprisingly, the acceleration
term is dominated by the velocity term, due to general relativity! (In some sense, Newton
modified by Einstein returns to Aristoteles.) As a consequence, an amazingly simple
approximate formula was proposed by Zeldovich [39]:
(0.9)
X(t, a) = a − t∇ϕ̃0 (a),
where ϕ̃0 is related to the behavior of the density field ρ(t, ·) at early stages
ρ(t, x) − 1
.
t↓0
t
The Zeldovich approximate formula suggests possible mass concentrations in finite time.
Indeed, denoting by Λ the largest eigenvalue of the Hessian matrix D 2 ϕ̃0 (a), for all a, we
see that, whenever Λ > 0, the map a → X(t, a) is no longer invertible at t = Λ−1 . Beyond
the concentration time, there are many possibilities of extending the formula and this is
still a controversial issue from the physical viewpoint. It depends very much on whether
or not we want to prevent interpenetration of particles. If we do so, we are naturally lead
to the model of adhesion dynamics, where particles merge after collisions, which is the
most possible dissipative behavior beyond concentrations. (See [5, 23, 24].) This issue
can be simply addressed in terms of nonlinear hyperbolic PDEs [22]. Indeed, given a
Zeldovich solution X defined by (0.9), let us introduce the field u(t, x) implicitly defined
by:
ρ(0, x) = 1, ∆ϕ̃0 = lim
(0.10)
u(t, X(t, a)) =
a − X(t, a)
= ∇ϕ̃0 (a),
t
3
as long as a → X(t, a) stays smooth and invertible. Then, we see that u solves the
multidimensional ”invisicid Burgers” equation
(0.11)
∂t u + (u · ∇)u = 0.
In one space dimension, if we want a global solution for all times, the monotonicity
condition ∂a X(t, a) ≥ 0 exactly corresponds to “Oleinik’s entropy condition” ∂x u ≤ 1/t,
which guarantees both global existence and uniqueness for solutions of the inviscid Burgers
equation (0.11), written in ”conservation form”
(0.12)
u2
∂t u + ∂x ( ) = 0.
2
Going back to the least action principle, it is remarkable that action (0.1) is (strictly)
convex in the variables (ρ, ρv, ϕ), while the differential constraints (0.2) are linear in
these variables. With classical tools of convex and functional analysis, Loeper [30] was
able to prove the existence of a unique minimizer in the class of probability measures
ρ(t, dx) which have no singular part with respect to the Lebesgue measure, i.e. without
concentrations, provided the data ρ0 and ρ1 are themselves concentration-free. [He also
rigorously derived optimality conditions (0.6).] This existence and uniqueness result is
quite remarkable. However, it provides only solutions to the Euler-Poisson system without concentrations. This is a major defect, since the initial value problem (as both ρ
and v are prescribed at initial time t0 ) is expected to have solutions with concentrations
appearing in finite time, for most initial conditions. So, minimizing action (0.1) under
constraints (0.2) is not a fully satisfactory way to recover solutions of the EP system from
the knowledge of ρ0 and ρ1 . The goal of the present paper is to investigate how the action
can be modified so that its minimizers are not necessarily concentration-free. A similar
problem, in the framework of adhesion-fragmentation processes, has been recently solved
by Wolansky [38]. (See also the pioneering work of Shnirelman [35] for sticky particles
and adhesion dynamics.) Our approach is different and more reminiscent of the recent
theory of self-dual lagrangians by Ghoussoub [27]. Unfortunately, it does not apply to
the desired Euler-Poisson system but rather to the modified system obtained by substituting the fully nonlinear Monge-Ampère equation ρ = det(I + tDx2 ϕ̃), for the Poisson
equation ρ = 1 + t∆ϕ̃ (in case of coefficients (0.3)). Notice that these systems agree
for solutions depending only on one spatial coordinate (i.e. with sheet structure) and
are asymptotically close for t << 1. Of course, changing the model is not a satisfactory
approach, without further justification. Our main argument is the following remarkable
property of the resulting Euler-Monge-Ampère system: it admits as exact solutions the
Zeldovich approximate solutions (0.9). As a secondary justification, let us recall that the
Euler-Poisson system is, after all, itself an approximation of the full Einstein equations
and it might be, from this viewpoint, equally good to use the Monge-Ampère equation
and the Poisson equation. [A similar situation occurs in fluid mechanics when comparing
the quasi-geostrophic and the semi-geostrophic approximations of the Euler equations for
ocean and atmosphere dynamics, as discussed in [21]. See also [18].] However, there will
be no attempt in the present paper to justify this last statement.
The structure of the paper is as follows:
4
In the first section, we introduce a family of infinite dimensional dynamical systems, in
a rather abstract (but straightforward) framework, requiring a limited amount of classical functional analysis (Hilbert spaces) and elementary convex analysis (sub-differential
calculus). Then, the ”Monge-Ampère” gravitation (MAG) model is introduced as a particular case. The model is first described in ”material variables” and then translated into
”Eulerian variables”, leading to a pressure-less Euler-Monge-Ampère system of PDEs.
In section 2, we return to the abstract framework and observe that the potential part of
the action has the very special property to be a squared distance function. This allows
a rewriting of the action as an exact square and we find as special minimizers all the
solutions of the gradient flow equation associated to the potential. [This special solutions
play more or less the same role as ”instantons” in Yang-Mills theory [27].] It turns out
that this gradient flow belongs to a very well-studied class of evolution equations, linked
to the theory of ”maximal monotone operators” [19]. This suggests a somewhat canonical
modification of the action.
In section 3, we adapt the previous discussion to Monge-Ampère gravitation and get a
modified action that takes into account concentrations. Then, we observe that the solutions of the gradient flow equation exactly are the Zeldovich approximate solutions (0.9)
of the Euler-Poisson system, which, in some sense, justifies the MAG model as a reasonable approximate model.
In section 4, we introduce a fully discrete algorithm to compute the EUR problem with
Monge-Ampère gravitation.
In section 5, we introduce a numerical scheme for the initial value problem and, finally,
in section 6, we provide numerical results in the very special case of one space variable.
1. The abstract framework
Let H be a (separable) Hilbert space H equipped with its norm denoted || · || and the
corresponding inner product ((·, ·)). We first consider the general dynamical system
d2 X
= (∇H Φ)[X],
dt2
where t → X(t) is valued in H, ∇H denotes the gradient operator in H, and Φ is a given
”potential” defined on H. (Observe that we do not follow the usual sign convention for
the potential, for notational convenience.) As well known, such a system admits a least
action principle, at least at a formal level. Indeed, for a curve t → X(t) valued in the
Hilbert space H, we may define its action between times t0 and t1 , t1 > t0 by:
Z t1
1 dX 2
(1.14)
A[t0 ,t1 ] [X] =
||
|| + Φ[X(t)] dt.
2 dt
t0
(1.13)
Then, the dynamical equation (1.13) can be seen as the formal optimality equation obtained by minimizing the action (1.14) as the end points X(t0 ) and X(t1 ) are fixed.
Next, we crucially assume the potential to be of form:
||X − s||2
; s ∈ S},
2
where S is a given bounded subset of H. Then, when it makes sense, X − ∇H Φ[X] is just
the closest point π[X] to X in the set S. (Clearly this definition is ambiguous whenever
(1.15)
Φ[X] = inf{
5
X has several distinct closest points, which may happen unless S is a convex set. In some
cases, X may have no closest point in S!) As a consequence, (1.13) formally means:
d2 X
= X − π[X],
dt2
where π[X] is the closest point to X on S. With this formulation, we can guess a large
class of explicit solutions. Indeed, let us assume that X(0) = X0 has a unique closest
point π[X0 ] = π0 on S. Then the linear (but not convex) combination of X0 and π0 given
by:
(1.16)
(1.17)
X(t) = π0 + et (X0 − π0 )
solves (1.16) as long as π0 stays the unique projection of X(t). Intuitively, X(t) gets
repelled from its initial position in the opposite direction of its closest point on S, keeping
for a while π0 as its closest point on S until a new point in S gets even closer. Whenever
S is a convex set, this repulsion mechanism provides an obvious global solution. Indeed,
all points contained in the infinite segment {π0 + r(X0 − π0 ), r ≥ 0} admits π0 as their
unique closest point on S. In the case of a non-convex set S, this is not true in general
and formula (1.17) is able to provide no more than a local solution. The situation is very
clear in the elementary case when S is the unit sphere in H. Then, 0 is the unique point
where Φ is not differentiable. We get as special solution
X(t) = r0−1 (1 + (r0 − 1)et )X0 ,
where X0 6= 0 and r0 = ||X0||. We see that, if r0 < 1, then the solution reaches 0 at time
T = − log(1 − r0 ) and its continuation beyond T gets ambiguous.
Miscellaneous mathematical remarks. 1) The potential Φ given by (1.15) is a smooth
perturbation of a Lipschitz concave function; indeed:
||X||2
− Π[X],
2
where Π is the Lipschitz convex functional defined by:
(1.18)
Φ[X] =
||s||2
; s ∈ S}.
2
A classical result of convex analysis [4] asserts that, for every Lipschitz convex function
defined on a Hilbert space, the set H̃ where the function is differentiable is always ”fat”
in the topological sense of Baire: namely H̃ is dense and contains a countable intersection
of dense open subsets of H [4]. In the particular case of Π, the set of differentiabliity
H̃ is contained in the set of all points X in H for which there is a unique closest point
s = π[X] on S. Thus, the potential Φ defined by (1.15) is everywhere differentiable on H̃
and its gradient in H is given by:
(1.19)
(1.20)
Π[X] = sup{((X, s)) −
∇H Φ[X] = X − π[X],
∀X ∈ H̃.
2) In the case when H is the finite-dimensional Hilbert space Rn , for such a potential (namely a smooth perturbation of a concave Lipschitz function), the dynamical
system (1.13) has a unique global solution for Lebesgue almost every initial condition
(X(0), X ′(0)) ∈ R2n and is, therefore, well-posed in the sense of Bouchut and Ambrosio
6
[8, 1]. To the best of our knowledge, there is no similar theory in infinite dimension and
the well-posedness of (1.13) is then a challenging open. (A somewhat related attempt is
the theory developed by Ambrosio and Gangbo for some infinite dimensional hamiltonian
systems [2]. See also [18, 25, 26].)
1.1. Monge-Ampère gravitation.
Definition. Since the dimension 3 does not matter in the definition of the MAG model, we
consider a smooth bounded closed domain D ⊂ Rd . We assume D to be of unit Lebesgue
measure. The MAG model is defined by choosing for H the Hilbert space of all Lebesgue
square-integrable maps from D to Rd ,
H = L2 (D, Rd),
(1.21)
and for S the subset of all Lebesgue measure-preserving maps s of D:
Z
Z
(1.22)
S={s∈H,
f (s(a))da =
f (a)da, ∀f ∈ C(Rd ) } .
D
D
In addition, with respect to the abstract framework, we input coefficients α, β given by
(0.3) and substitute
d
dX
(1.23)
β −2 (t) (α2 (t)
) = (∇H Φ)[X] = X − π[X],
dt
dt
for (1.13).
Eulerian formulation. We now want to write the MAG model in ”Eulerian coordinates”.
Given a solution t → X(t) ∈ H, we introduce the pair of measures (ρ, ρv), respectively
valued in R and Rd and defined by:
Z
ρ(t, dx) =
δ(x − X(t, a))da,
D
Z
(1.24)
v(t, x)ρ(t, dx) =
∂t X(t, a)δ(x − X(t, a))da.
D
Using the tools of ”optimal transportation theory” [11, 12, 37] (cf. Appendix), we (formally) get for (ρ, v) the following system of partial differential equations:
(1.25)
∂t ρ + ∇ · (ρv) = 0,
∂t (α2 ρv) + ∇ · (α2 ρv ⊗ v) = −β 2 ρ∇ϕ ,
ρ = det(I + Dx2 ϕ),
where (ρ, v, ϕ)(t, x) ∈ R+ × Rd × R, (t, x) ∈ R × Rd ,
I = (δij ), Dx2 = (∂x2i xj ).
Thus our model can be seen as a nonlinear correction of the classical pressure-less EulerPoisson system (EP), obtained by substituting the fully nonlinear Monge-Ampère equation
(1.26)
ρ = det(I + Dx2 ϕ).
for the linear Poisson equation
(1.27)
ρ = 1 + ∆ϕ
7
For this reason, we call Monge-Ampère gravitation (MAG) the model defined by either
(1.16,1.21,1.22) or (1.25). Notice that the MAG model coincides with the conventional
Euler-Poisson model in dimension d = 1.
2. Modified action in the abstract framework
In this section, we go back to the abstract framework of a potential Φ defined as
the squared distance to a bounded subset S of a general Hilbert space H, according to
(1.16,1.18,1.19).
2.1. The self-dual form of the action. Since potential Φ is a squared distance to some
subset S inside H, it solves, at least formally, the stationary Hamilton-Jacobi equation:
||∇H Φ||2
,
2
where ∇H denotes the gradient operator in H. This suggest to rewrite, at least formally,
the action (1.14) as
Z t1
1 dX 2 ||∇H Φ[X(t)]||2
||
|| +
dt
2 dt
2
t0
Z t1
1 dX
dX
(2.29)
=
||
− ∇H Φ[X(t)]||2 + ((
, ∇H Φ[X(t)])) dt
2 dt
dt
t0
Z t1
1 dX
= Φ[X(t1 )] − Φ[X(t0 )] +
||
− ∇H Φ[X(t)]||2 dt.
2
dt
t0
(2.28)
Φ=
Under this ”self-dual” form (see [27] for a systematic study of ”self-dual lagrangians”), it
is obvious that any solution of
dX
= (∇H Φ)[X] = X − (∇H Π)[X]
dt
is always a minimizer of the action as X(t0 ) and X(t1 ) are fixed (just like instantons in
euclidean Yang-Mills theory, cf.[27]).
(2.30)
2.2. A gradient flow equation. As already mentioned, in spite of the rather nice structure (1.18) of Φ, as a quadratic perturbation of a convex Lipschitz function, the corresponding second-order equation (1.16) is not so well understood. In sharp contrast, the
first-order equation (2.30) is a standard ”gradient flow ” equation (GF), that can be solved
by classical ”maximal monotone operator” theory [19].
In the framework of maximal monotone operator theory, equation (2.30) is usually written
as a sub-differential inclusion:
dX
(2.31)
−
+ X ∈ ∂Π[X],
dt
which is well-posed in H since Π is Lipschitz and convex. Here, we use standard notations
of convex analysis, for which ∂ denotes the sub-differential of a convex function [19]:
(2.32)
∂Π[X] = {Z ∈ H; Π[Y ] ≥ Π[X] + ((Z, Y − X)), ∀Y ∈ H}.
8
A remarkable property [19] of each solution X(t) ∈ H is to be not only a Lipschitz
continuous function of t but also right-differentiable at each t with
(2.33)
−
dX(t + 0)
+ X(t) = d0 Π[X(t)],
dt
∀t
where d0 Π[X], following [3], denotes the element of ∂Π[X] with minimal norm (which is
uniquely defined):
(2.34)
||d0Π[X]|| = min{||s|| ; s ∈ ∂Π[X]}.
Finally, notice that X(t) is a locally Lipschitz function of t with values in the separable
Hilbert space H, X(t) is therefore almost everywhere differentiable in t by Rademacher
theorem. Since X(t) is right-differentiable everywhere, we conclude that:
(2.35)
−
dX
+ X(t) = d0 Π[X(t)],
dt
holds true both in the almost everywhere sense and in the sense of distributions.
2.3. Proposals for a modified action. Our main point is now to introduce a modif ied
action. There are two possible ways to do it. First, we may introduce the modified
potential Φ̃:
(2.36)
1
Φ̃[X] = ||X − d0 Π[X]||2
2
and the corresponding modified action
Z t1
Z t1
1 dX 2 1
1 dX 2
||
|| + Φ̃[X(t)] dt =
||
|| + ||X − d0 Π[X]||2 dt.
(2.37) Ã[t0 ,t1 ] [X] =
2 dt
2 dt
2
t0
t0
Alternately, sticking more closely to the self-dual formulation, we may directly modify
the action by setting
Z t1
1 dX
(2.38)
Â[t0 ,t1 ] [X] =
||
− X + d0 Π[X]||2 dt.
2
dt
t0
It is not clear to us that these modified actions coincide (up to boundary terms). Nevertheless, we will take the second option, mostly for numerical purposes, because it leads
to simpler algorithms.
3. Application to the EUR problem with Monge-Ampère gravitation
We now address the Monge-Ampère gravitation (MAG) model and the EUR problem.
This means, with respect to the abstract framework, that H and S are now defined by
(1.21,1.22) and (1.23) substitutes for (1.13).
9
3.1. A modified action for the EUR problem. In order to take into account coefficients (α, β) (given by (0.3)), we first rewrite the action as:
Z t1
dX 2
|| + β(t)2 ||∇H Φ[X(t)]||2 dt .
α(t)2 ||
(3.39)
A=
dt
t0
As in the homogeneous case α = β = 1, we keep in mind that
1
||∇H Φ[X(t)]||2 = Φ[X(t)]|
2
and look at the cross-term:
Z t1
Z t1
dX
d
α(t)β(t)((
α(t)β(t) (Φ[X(t)])dt
J=
, ∇H Φ[X(t)]))dt =
dt
dt
t0
t0
By integration by part, we get
Z
J − α(t1 )β(t1 )Φ[X(t1 )] + α(t0 )β(t0 )Φ[X(t0 )] = −
1 t1
d
λ
||∇H Φ[X(t)]||2 (α(t)β(t))dt = −
2 t0
dt
2
provided we assume
=−
Z
Z
t1
Φ[X(t)]
t0
d
(α(t)β(t))dt
dt
t1
t0
β(t)2 ||∇H Φ[X(t)]||2 ,
d
(α(t)β(t)) = λβ(t)2 ,
dt
(3.40)
√
for some constant λ, which is is consistent with data (0.3) if we choose λ = 1/ 6. From
this calculation of the cross-term J, we deduce that the action A defined by (3.39) can
be written:
Z t1
dX
− µβ(t)∇H Φ[X(t)]||2 dt .
||α(t)
A = BT +
dt
t0
2
where BT is a boundary term√depending only
p on X(t1 ) and X(t0 ), provided µ + µλ = 1.
For data (0.3), we get λ = 1/ 6 and µ = 2/3. Therefore, all solutions of the gradientflow equation
dX
= µβ(t)∇H Φ[X(t)]
dt
automatically are minimizers of the action (3.39). For data (0.3), this gradient-flow equation reduces to:
dX
(3.42)
t
= ∇H Φ[X(t)] = X(t) − ∇H Π[X(t)].
dt
The gradient-flow equation should be understood in the more precise sense:
(3.41)
α(t)
dX(t + 0)
= X(t) − d0 Π[X(t)],
dt
which takes concentration into account, globally in time. In some sense, formulation
(3.43) not only allows concentrations but guarantees the largest possible dissipation of
kinetic energy during the concentration process (which is of course questionable from the
(3.43)
t
10
physical viewpoint.) Accordingly, we suggest, for the EUR problem with Monge-Ampère
gravitation, the following modified action:
Z t1
dX
− X + d0 Π[X]||2 dt.
t−1/2 ||t
(3.44)
 =
dt
t0
3.2. Zeldovich solutions. Special solutions of (3.42) can be obtained thanks to the
concept of ”rearrangements with convex potential” as follows. By definition, the MAG
model relies on the set S of all Lebesgue measure-preserving maps (1.22). This set contains
the identity map Id as an obvious element. The set K ⊂ H of all points X which admits
Id as a closest point on S plays a crucial role. It can be characterized (cf. Appendix
on optimal transportation theory), as the convex cone of all maps X ∈ H with a convex
potential, which means that there is a convex function ψ defined on Rd and valued in
] − ∞, +∞] which is almost everywhere differentiable on D with ∇ψ(a) = X(a), a.e. on
D. It turns out that any map X ∈ H has a unique rearrangement X ♯ in K, which means
Z
Z
♯
δ(x − X (a))da =
δ(x − X(a))da
D
D
(cf. Appendix).
Therefore, special solutions of (3.42) can be obtained, by looking for solutions X(t) valued
in the convex cone K of all maps with convex potential. Indeed, for such solutions, we
have:
∇H Φ[X(t)] = X(t) − ∇H Π[X(t)] = X(t) − π[X(t)] = X(t) − Id,
and (3.42) reduces to the linear ODE
dX
= X − Id
dt
as long as X(t) belongs to K, i.e. X(t, a) = ∇ψ(t, a), with ψ(t, a) convex in a. This leads
to the explicit formula:
(3.45)
(3.46)
t
X(t, a) = ∇ψ(t, a) = a +
t
t
(∇ψ(t0 , a) − a) = a + (X(t0 , a) − a) ,
t0
t0
as long as ψ stays convex in a. This exactly coincides with Zeldovich formula (0.9) discussed in the introduction. Remarkably enough, for Monge-Ampère gravitation, Zeldovich
approximation (0.9) is just exact!
3.3. Modified action in one space dimension. Let us focus on the one space dimension case when: D = [−1/2, 1/2]. Then, the modified potential Φ̃ can be explicitly
computed in the case of a piecewise smooth map Y valued in K. Indeed, in one space
dimension, maps in K, with convex potential are just increasing maps. So, there is a
finite number of plateaux [aj , bj ] on which Y is constant with values Yj and outside of
which Y is a piecewise smooth strictly increasing function.
Notice that the corresponding image-measure ρ(dx) defined by
Z
ρ(dx) =
δ(x − Y (a))da
D
11
has a singular part ρs given by:
ρs (dx) =
X
j
(bj − aj )δ(x − Yj ).
Then d0 Π[Y ] (the element of the sub-differential ∂Π[Y ] with minimal L2 norm) coincides
with the identity map outside of the plateaux and takes value (aj + bj )/2 inside [aj , bj ].
After elementary calculations, we find
X Z bj
aj + bj 2
0
2
2
2
) da
||Y − d Π[Y ]|| = ||Y || − 2((Y, Id)) + ||Id|| −
(a −
2
a
j
j
= ||Y − Id||2 −
1 X
(bj − aj )3 .
12 j
Here we very clearly see the discrepancy between the original potential Φ and the modified
potential Φ̃:
1 X
(3.47)
Φ̃[Y ] = Φ[Y ] −
(bj − aj )3 .
24 j
Remark. Specialists of nonlinear hyperbolic conservation laws will recognize in the second
term of this expression the very expression of the so-called ”entropy production” term for
the inviscid Burgers equation (0.12), written in material coordinates [6, 22].
3.4. Eulerian formulation of the gradient flow equation. The gradient flow equation (3.41) has an Eulerian version. Indeed, the corresponding measures (ρ, ρv), defined
by (1.24), are (formal) solutions of the following system of PDE:
(3.48)
∂t ρ − ∇ · (ρ∇ϕ̃) = 0, ρ = det(I + tDx2 ϕ̃),
where ϕ̃(t, x) = ϕ(t, x)/t. This model can be seen as a fully nonlinear counterpart of various models popular in biology (chemotaxis) or astronomy, involving the Poisson equation
-or other linear equations involving a singular Green function- rather than the MongeAmpère equation. A common feature of all these models is their ability at describing
concentration phenomena [29, 28, 33, 20].
4. The discrete EUR problem with Monge-Ampère gravitation
4.1. A time-discrete scheme for the gradient flow equation. In view of numerical
calculations, our first step is to get a time-discrete version of the modified action. Instead
of directly getting a discrete version of (3.44), it seems wiser to us to start from a timediscrete version of the gradient flow equation (3.42).
A natural candidate is:
(4.49)
Xn+1 = Xn (1 + θn ) − ∇H Π[Xn ]θn + ηn
where Xn is an approximation of X(t) at the nth time-step Tn , for n = 0, · · ·, N, T0 = t0 ,
− 1 ↓ 0. Here ηn is a small perturbation added to the discrete
TN = t1 , with θn = TTn+1
n
solution so that, for every n, Xn is a point of differentiablity of Π. Thus, ∇H Π[Xn ] is well
defined and is also the closest point π[Xn ] to Xn in S. Indeed, as a smooth perturbation
of a Lipschitz concave function on H, Π is differentiable on a ”fat” dense subset H̃ of
H (i.e. containing a countable intersection of dense open sets). Thus, we may choose a
12
perturbation term ηn , arbitrarily small, so that Xn falls in the ”good” set H̃ where ∇H Π
is well defined. By doing so, we do not generate a big error. Indeed, we keep control on
the cumulated error thanks to the following stability estimate for two distinct solution Xn ,
X̃n of (4.49) (where we neglect the perturbation terms ηn , η̃n for notational simplicity),
||Xn+1 − X̃n+1 ||2 ≤ (1 + θn )2 ||Xn − X̃n ||2 + c0 θn2 ,
(4.50)
where c0 is the squared diameter of S. [Indeed, we get form (4.49)
||Xn+1 − X̃n+1 ||2 = (1 + θn )2 ||Xn − X̃n ||2 − 2(1 + θn )θn ((Xn − X̃n , ∇H Π[Xn ] − ∇H Π[X̃n ]))
+θn2 ||π[Xn ] − π[X̃n ]||2
and observe that the second term in the right-hand side is less than zero since Π is convex,
and the third one is dominated by c0 θn2 .]
As a matter of fact, this stability estimate is also essentially sufficient to prove the convergence of the scheme as θn ↓ 0 to the continuous model (3.43), for the uniform convergence
in time with respect to the strong topology of H. (See [13, 14] for examples of similar
results for various nonlinear hyperbolic conservation laws.) Notice that concentration
phenomena, which are present at the continuous level, are correctly taken into account
by the time discrete scheme, in spite of the fact that the discrete scheme never involves
the computation of d0 Π, which is a big advantage in practice!
4.2. The time-discrete EUR problem with Monge-Ampère gravitation. From
the time-discrete scheme (4.49) for the gradient-flow equation, we define a time discrete
version of modified action (3.44) just by setting:
N
−1
X
(4.51)
n=0
rn ||Xn+1 − Xn (1 + θn ) + π[Xn ]θn ||2 ,
3/2
Tn
Tn+1 −Tn
. Nevertheless, minimizing the time-discrete action (4.51), as both X0
with rn =
and XN are fixed, is not really consistent with the original EUR problem, where the data
are not X0 and XN but rather the corresponding probability measures ρ0 and ρN defined
by:
Z
Z
ρ0 (dx) =
δ(x − X0 (a))da, ρN (dx) =
δ(x − XN (a))da.
D
D
So there is a big loss of information (since the same probability measure can be generated
by a continuum of maps). This problem can be addressed in terms of rearrangements
with convex potentials. As a matter of fact, fixing ρ0 and ρN is equivalent to fixing the
rearrangements with convex potentials X0♯ and XN♯ , rather than X0 and XN themselves.
It is very fortunate that, one can rewrite the discrete scheme (4.49) as a self-consistent
scheme for the rearrangement Yn = Xn♯ with convex potential. Indeed, let us assume,
for simplicity, that, at each n, the solution of the scheme Xn has a polar factorization
Xn = Yn ◦ sn (cf. Appendix), where Yn = Xn♯ ∈ K is the unique rearrangement with
convex potential of Xn and sn = π[Xn ] ∈ S is the closest point in S to Xn . Then, we can
rewrite (4.49) as:
Yn+1 ◦ sn+1 = (Yn (1 + θn ) − θn Id) ◦ sn .
13
But, this implies that Yn+1 is the unique rearrangement of Yn (1 + θn ) −θn Id with a convex
potential. In other words, we have a well defined self-consistent scheme for Yn ∈ K,
namely:
Yn+1 = (Yn (1 + θn ) − θn Id)♯ .
(4.52)
Accordingly, the time discrete EUR problem with Monge-Ampère gravitation can be seen,
in ”polar coordinates” (Yn , sn ) ∈ K × S, as the minimization of
N
−1
X
(4.53)
n=0
with rn =
3/2
Tn
,
Tn+1 −Tn
rn ||Yn+1 ◦ sn+1 − (Yn (1 + θn ) − θn Id) ◦ sn ||2 ,
as Y0 and YN are fixed in K.
Following ”optimal transport” theory, we may introduce on H the quadratic MongeKantorovich (MK2) (or ”Wasserstein”) distance,
dM K 2 (X, X̃) = inf{||X ◦ s − X̃ ◦ s̃||, s, s̃ ∈ S },
(4.54)
which is nothing but the quotient distance in H with respect to the action of the semigroup S. Then the discrete EUR problem is just the minimization in Yn ∈ K of
N
−1
X
(4.55)
n=0
rn dM K 2 (Yn+1 , Yn (1 + θn ) − θn Id)2 ,
3/2
Tn
, as Y0 and YN are fixed in K.
with rn = Tn+1
−Tn
(Equivalently, we could work on the so-called ”Wasserstein” or ”MK2” space as, for
instance, in [34, 3, 2].)
4.3. The fully discrete least action principle. Let us now introduce a fully discrete
scheme, for which not only the time variable but also the space variable is discrete. The
domain is divided into L disjoints subdomains Di of Lebesgue measure 1/L, for i = 1, ···, L,
with barycenter ai and vanishing diameter as L → ∞. In our abstract framework, it is
enough to substitute for the spatial domain D, the discrete set {ai , i = 1, · · ·, L }.
Accordingly, H can be seen as the euclidean space (Rd )L of all finite sequences of L
points in Rd {X = (Xi ∈ Rd )i=1,L } with the natural euclidean norm || · || induced by Rd .
Meanwhile the set S can be viewed as the set of all permutations s of the L first integers
and K is the corresponding cone of all sequences Yi such that
X
Yi · (ai − asi ) ≥ 0,
i
for all permutations s. In one space dimension, K is just the convex cone of all increasing
sequences of L real numbers.
The time-discrete EUR problem (4.53) makes sense at the fully discrete level without
modification. In this discrete setting, S is a group (which is untrue at the continuous
level) and each s can be inverted in S. Thanks to the group property of S and the invariance of || · || with respect to S, the minimization problem can be further reduced to the
14
minimization of
(4.56)
N
−1
X
n=0
rn ||Yn+1 − (Yn (1 + θn ) − θn Id) ◦ σn+1 ||2 ,
in Yn ∈ K, σn ∈ S, as Y0 and YN are fixed in K (just by setting σn+1 = sn ◦ s−1
n+1 ∈ S).
To solve this minimization problem, a crude strategy is to use Gauss-Seidel type iterations.
We denote by (Ynk , σnk ) the approximation of (Yn , σn ) at iteration k and time step n. Let
us fix k and n. To get the updated values σnk+1 and Ynk+1 , we inductively suppose that
j
we already know (Ymj , , σm
) for all m if j ≤ k and for all m < n if j = k + 1. Then, we
perform the two following steps:
i) First step: we get s = σnk+1 by solving the combinatorial optimization problem
(4.57)
k
inf ||Yn+1
− (Ynk (1 + θn ) − θn Id) ◦ s||.
s∈S
This step is particularly simple in one space dimension and just amounts to sorting in
increasing order the finite sequence (Ynk (1 + θn ) − θn Id)i, i = 1, · · ·, L. It is much more
challenging in higher dimensions. The best known optimization methods need 0(L3 )
elementary operations, which is not satisfactory (see a related discussion in [17]).
ii) Second step: we get Y = Ynk+1 by minimizing in Y ∈ K:
k
k
k+1
rn ||Yn+1
− (Y (1 + θn ) − θn Id) ◦ σn+1
||2 + rn−1 ||Y − (Yn−1
(1 + θn−1 ) − θn−1 Id) ◦ σnk+1||2 ,
where the first term can also be written
k
k
rn ||Yn+1
◦ (σn+1
)−1 − (Y (1 + θn ) − θn Id)||2,
k
using the inverse permutation (σn+1
)−1 and the invariance of || · || with respect to permutations. After reorganizing squares, we see that Y is just the least-square projection
H → K of:
rn (1 + θn )W + rn−1 Z
V =
,
rn (1 + θn )2 + rn−1
k
k
k+1
W = Yn+1
◦ (σn+1
)−1 + θn Id, Z = (Yn−1
(1 + θn−1 ) − θn−1 Id) ◦ σnk+1 .
So, we have obtained an effective algorithm. It is particularly simple in one space dimension (and much more challenging in higher dimensions!). Let us observe that, in one space
dimension, computing the least-square projection Y = PK [V ] is different from sorting the
sequence V in increasing order. However, still in one space dimension, this projection can
be approximately computed after a sequence of sorting steps, according to the asymptotic
formula (which is a byproduct of the ”transport-collapse method” [10]):
1
M
(4.58)
PK [V ] = lim VMM , VmM = (Vm−1
+ V )♯ , V0M = 0, m = 1, · · ·M.
M →∞
M
In practice, we already get a good accuracy for moderate values of M (say M = 10).
5. Solution of the initial value problem
In order to validate the reconstruction scheme, we would like to solve the initial value
problem (IVP) consistently with the modified least action problem, and get a discrete
scheme for the IVP. Ideally, such a scheme should be derived directly from the modified
discrete least action principle. Unfortunately, we have not been able to do so, and we are
15
just going to suggest a simple scheme for the IVP which seems, in practice, consistent
with the modified action, at least in one space dimension.
5.1. A time-discrete scheme for the IVP. Our idea to get a time-discrete solution
of the IVP is to alternate the solution of the linear ODE
(5.59)
d 2
(α (t)V ) = β 2 (t)(Y − Id),
dt
dY
= V,
dt
Y (Tn ) = Yn , V (Tn ) = Vn ,
with coefficients (α, β) given by (0.3), on each time interval [Tn , Tn+1 [ and the rearrangement of the result at time step Tn+1 :
Yn+1 = Y (Tn+1 )♯ , Vn+1 = V (Tn+1 ).
Using a plain explicit discretization of (5.59), we get:
(5.60)
Yn+1 = (Yn + (Tn+1 − Tn )Vn )♯ ,
α2 (Tn+1 )Vn+1 = α2 (Tn )Vn + β 2 (Tn )(Tn+1 − Tn )(Yn − Id)
The convergence analysis of this time-discrete scheme can be done in two different ways.
5.2. The multidimensional case. In the multidimensional case, our strategy for the
convergence analysis of scheme (5.60) is inspired by our recent work [15], where a similar
scheme is analyzed. We essentially use the fact that all maps with convex potential are
of locally bounded variations, which provides enough compactness with respect to space
variables. Time compactness is, as usual, directly obtained from the evolution scheme.
We notice that V can be easily integrated out from Y by ODE (5.59).
Theorem 5.1. For every fixed initial condition (Y0 , V0 ) ∈ Y ×H, the approximate solution
(Yn ) admits at least a limit t → Y (t) ∈ K valued in C 0 ([t0 , +∞[, H) as the time step goes
to zero. This limit satisfies:
Z t
2
2
V (t, a)α (t) = V0 (a)α (t0 ) +
β 2 (τ )(Y (τ, a) − a)dτ,
(5.61)
Z
Z t0
d
f (Y (t, a))da = (∇f )(Y (t, a)) · V (t, a)da,
dt D
D
for all C 1 function f on Rd (with |∇f | growing at most linearly at infinity).
R
Notice that, since Y is valued in K, the knowledge of ”observables” D f (Y (t, a))da
for all test-functions f is enough to determine Y (t) which makes formulation (5.61) selfconsistent. (However this does not guarantee uniqueness of solutions to the IVP.) So, we
have a proposal to solve the IVP, and a corresponding discrete scheme, but we are not
able to prove that formulation (5.61) is actually consistent with our modified least action
principle.
5.3. The one-dimensional case. In the special case of one space variable, we get a much
more precise information, following the analysis developed in [13] for similar problems (see
also [14]):
16
Theorem 5.2. For every fixed initial condition (Y0 , V0 ) ∈ Y × H, as the time step goes
to zero, the approximate solution (Yn ) converges to the unique solution t → Y (t) ∈ K,
valued in C 0 ([t0 , +∞, H), of the mixed integral-sub-differential system:
(5.62)
V (t, a)α2 (t) = V0 (a)α2 (t0 ) +
Z
−∂t Y + V ∈ ∂Θ[Y ],
t
t0
β 2 (τ )(Y (τ, a) − a)dτ,
where Θ[Y ] = 0 whenever Y = Y (t, a) is monotonically increasing in a and Θ[Y ] = +∞
otherwise.
System (5.62) is well-posed in the L2 sense and can be shown (as in [13]) to be the limit
(in the sense of maximal monotone operator theory) as ǫ ↓ 0 of the perturbed system
(5.63)
−∂t Y + V = −ǫ∂a (log(∂a Y )), ∂t (α2 (t)V ) = β 2 (t)(Y − Id),
which, in Eulerian variables (1.24), reduces to:
∂t ρ + ∂x (ρv) = 0,
∂t (α2 (t)ρv) + ∂x (α2 (t)ρv 2 ) = −β 2 (t)ρ∂x ϕ + ǫ∂xx v,
(5.64)
ρ = 1 + ∂x2 ϕ,
and is just a pressure-less Navier-Stokes-Poisson system with vanishing viscosity (as in
[9, 36]). As mentioned above, an interesting open question is to show that this approach
(vanishing viscosity (5.64), subdifferential formulation (5.62) or scheme (5.60)) is actually
consistent with the modified least action principle!
6. Numerical simulations in one space dimension
Our data are
t0 = 1/2, t1 = 5/2, N = 60, L = 51, ai = −1 + (2i − 1)/L,
i = 1, · · ·L,
Y0 = X0♯ , (X0 )i = ai ωi , ,
where ωi is a random number uniformly distributed between 1 and 2. Thus, Y0 looks like
a devil’s staircase.
Concerning the final data YN ∈ K, either:
(Case 1) the associate probability ρN is the barycenter of four Dirac’s measures:
δ(x + 0.7) + 4 δ(x − 0.2) + 3 δ(x − 0.9) + δ(x − 1.1)
.
9
or (Case 2) YN is the solution at time t1 of the initial value problem generated by the
discrete gradient flow equation starting from Y0 = X0♯ at time t0 .
ρN (dx) =
In our plots, we draw the trajectories of the 51 particles during the 60 time steps of
the time interval (the vertical axis corresponding to time and the horizontal one to space).
Case 1: we first plot the reconstructed solution (fig. 1). Then, with the reconstructed
initial velocity we solve the initial value problem for the MAG equations with scheme
(5.60) and plot the result (fig.2). We observe a nearly perfect match between figures 1
and 2.
17
Case 2: we first solve the IVP for the gradient flow equation with scheme (5.60) (fig.
3). Then, we reconstruct the solution from the initial and final data of the gradient flow
solution (fig 4). (Here we observe some limited discrepancy.) Finally, with the reconstructed initial velocity we solve the initial value problem for the MAG equations (fig.5),
again with scheme (5.60) and get a nearly perfect match.
18
2.5
2
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 1. Case 1/reconstruction
2.5
2
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 2. Case 1/initial value problem (IVP) after reconstruction
7. Discussion
We have revisited the early universe reconstruction problem and suggested a modification of classical Newton gravitation by what we called Monge-Ampère gravitation. The
main drawback of our approach is the lack of physical justification for such a modification.
The main mathematical advantage is the obtention of a modified least action principle
in which we can easily include mass concentration effects in an almost canonical way,
using ideas from gradient flow theory. In addition, the well-known Zeldovich approximate
solutions turn out to be exact solutions of the modified model, which provides an indirect
validation of the model as a reasonable approximation for the early universe reconstruction (EUR) problem. According to these ideas, an algorithm has been designed in the 1D
case. Our plan for the future includes: i) analysis of the initial value problem, consistently
with the modified least action principle; ii) design of an efficient multidimensional algorithm; iii) study of the relative accuracy of the Newton and Monge-Ampère gravitation
models with respect to general relativity.
19
2.5
2
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 3. Case 2/gradient flow solution -IVP before reconstruction
2.5
2
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 4. Case 2/gradient flow solution -reconstruction
2.5
2
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 5. Case 2/gradient flow solution -IVP after reconstruction
20
8. Appendix
8.1. Some useful results from optimal transport theory. The set S defined by
(1.22) has a semi-group structure for the composition rule and has the identity map Id
as neutral element. It is, in some sense, in duality with its ”polar cone” K ⊂ H:
(8.65)
K = {Y ∈ H ; ((Y, Id − s)) ≥ 0, ∀s ∈ S} .
Let us recall few basic results of optimal transport theory [11, 12, 37] concerning S and
K. First, the set K can be characterized as the closed convex cone of all maps Y with a
convex potential, which means that there is a convex function ψ defined on Rd and valued
in ] − ∞, +∞] which is almost everywhere differentiable on D with ∇ψ(x) = Y (x), a.e.
on D.
Next, every map admits a unique rearrangement in K. More precisely
Theorem 8.1. ([11]) Every X ∈ H admits a unique ”rearrangement” X ♯ in K, which
means:
Z
Z
♯
f (X (a))da =
f (X(a))da, ∀f ∈ C(Rd ), sup |f (x)|(1 + |x|2 )−1 < +∞.
D
x
D
In addition, X → X ♯ is continuous in H (for the strong topology).
Moreover, there is a ”polar factorization” of the Hilbert space H by S and K. More
precisely:
Theorem
R 8.2. ([11]) Let X ∈ H be a non degenerate map, in the sense that the measure
ρ(dx) = D δ(x − X(a))da has no singular part with respect to the Lebesgue measure.
Then X admits a unique ”polar factorization”
(8.66)
X = Y ◦ s,
Y ∈ K, s ∈ S.
In addition, the second factor s is characterized as the unique closest point π[X] to X in
S and can be written
(8.67)
π[X] = ∇ψ ◦ X,
where ∇ψ is the unique map T : Rd → D with convex Lipschitz potential such that the
Lebesgue measure restricted to D is the image of ρ by T :
Z
Z
(8.68)
f (∇ψ(x))ρ(dx) =
f (a)da, ∀f ∈ C(Rd ).
Rd
D
Let us finally observe as in [11, 12] that (8.68) can be seen as a ”weak formulation” (not
in the sense of distributions!) of the Monge-Ampère problem on Rd with range condition:
(8.69)
ρ = det(Dx2 ψ),
(∇ψ)(Rd ) = D.
8.2. Formal derivation of the MAG system. Let us first notice that the measures
(ρ, ρv) defined by (1.24) automatically satisfy the mass conservation equation in the sense
of distributions:
(8.70)
∂t ρ + ∇ · (ρv) = 0.
21
Next, let us assume that X is non degenerate at time t (which exactly means that ρ(t, dx)
has no singular part with respect to the Lebsegue measure).
Then
∇H Φ[X(t)] = X(t) − π[X(t)] = (Id − ∇ψ(t, ·)) ◦ X(t),
as in Theorem 8.1 and we can write the MAG equation (1.23):
d 2 dX
(α (t)
) = (Id − ∇ψ) ◦ X,
dt
dt
where ∇ψ(t, ·) is the unique ”weak solution” of the Monge-Ampère problem (8.69).
In terms of (ρ, v), (8.71) formally means
(8.71)
(8.72)
β −2 (t)
∂t (α2 ρv) + ∇ · (α2 ρv ⊗ v) = −ρβ 2 (t)(x − ∇ψ),
which, combined with (8.69), coincides with the Eulerian formulation (1.25) of the MAG
model, with ϕ(t, x) = ψ(t, x) − |x|2 /2. Of course, this derivation is purely formal: not
only concentrations have not been taken into account, but the derivation of (8.72) from
(8.71) is far from being legitimate.
Acknowledgments. The author acknowledges the support of ANR contract OTARIE ANR07-BLAN-0235. He warmly thanks the hospitality of the Forschunginstitut Mathematik
of the ETHZ that he visited in the fall of 2009.
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CNRS, FR 2800, Université de Nice Sophia, Parc Valrose, FR06108 Nice, France
E-mail address: [email protected]